Number Theory Portal

1Overview

Number theory studies divisibility, greatest common divisors, prime factorization, and remainders of integers. This portal proceeds from foundational definitions to computational methods and theorems, then organizes applications to congruences and approximation according to their prerequisites.

2Common foundation

First define divisors, multiples, quotients, and remainders, and use congruences to describe relations among remainders. Then derive the Euclidean algorithm and Bézout's identity, solve linear Diophantine equations, and study uniqueness of prime factorization.

3Congruence path

The Chinese remainder theorem combines several congruence conditions with pairwise coprime moduli into one residue class.

4Continued-fraction path

A continued fraction records the quotients of the Euclidean algorithm iteratively. It terminates for rational numbers and continues indefinitely for irrational numbers. Its convergents provide rational approximations to real numbers.

5Application to calendar arithmetic

Calendar arithmetic applies the Chinese remainder theorem to coprime periods; for non-coprime periods, it checks compatibility and uses the least common multiple. Its astronomical approximations use continued fractions introduced in the preceding lecture.

6Advanced connections

Infinite descent proves the impossibility of certain Diophantine equations. Abstract algebra interprets congruence as equality of residue classes and formalizes modular arithmetic as operations on those classes. Both provide advanced perspectives on number theory.